For what value of k is the polynomial p(x) = 2x3 – kx2 + 3x + 10 exactly divisible by (x + 2)?

Foundation

Mathematics Foundation

Factor theorem

Question

For what value of k is the polynomial p(x) = 2x³ – kx² + 3x + 10 exactly divisible by (x + 2)?

$\frac{1}{3}$

– 3

$- \frac{1}{3}$

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For a polynomial p(x) to be divisible by (x + 2), then x = – 2 must be a root. This means p(– 2) must equal 0.

Substitute x = – 2 into p(x) and set it equal to zero:

2(− 2)³ − k(− 2)² + 3(− 2) + 10 = 0

Simplify the equation:

2(− 8) − k(4) − 6 + 10 = 0

− 16 − 4k + 4 = 0

− 4k − 12 = 0

Solve for k: − 4k = 12 → k = − 3

Final Answer: – 3